Scaling Theorem

The scaling theorem (or similarity theorem) provides
that if you horizontally ``stretch'' a signal by the factor
in the time domain, you ``squeeze'' and amplify its Fourier transform
by the same factor in the frequency domain. This is an important
general Fourier duality relationship.

Theorem: For all continuous-time functions
possessing a Fourier
transform,

(B.9)

where

(B.10)

and
is any nonzero real number (the abscissa stretch factor).
A more commonly used notation is the following:

(B.11)

Proof:
Taking the Fourier transform of the stretched signal gives

The absolute value appears above because, when
,
, which brings out a minus sign in front of the
integral from
to
.